×

Categorical aspects of toric topology. (English) Zbl 1149.55014

Harada, Megumi (ed.) et al., Toric topology. International conference, Osaka, Japan, May 28–June 3, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4486-1/pbk). Contemporary Mathematics 460, 293-322 (2008).
The aim of this paper is to recall aspects of category theory with influence on the toric topology development by surveying recent examples and applications. The primary objects of study are derived from the well-behaved actions of the \(n\)-dimensional torus \(T^n\) on a topological space, and lie in a variety of geometric and algebraic categories. Each of the orbit spaces is equipped with a natural combinatorial structure, which encodes the distribution of isotropy subgroups and is determined by a finite simplicial complex \(K\). In this context, topological and homotopy theoretic invariants of toric spaces are described and evaluated in terms of combinatorial data associated to \(K\). Many familiar invariants of toric spaces depend only on their homotopy type, yet homotopy equivalences do not interact well with colimits. In addition, many of the functors one wishes to apply to toric spaces do not respect colimits. To develop these themes, the authors work with Quillen model categories [D. G. Quillen, Homotopical algebra. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0168.20903)] and interpret certain homotopy colimits as algebraic models of toric spaces.
For the entire collection see [Zbl 1140.57001].

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
57R91 Equivariant algebraic topology of manifolds
57N65 Algebraic topology of manifolds
55P35 Loop spaces

Citations:

Zbl 0168.20903
PDFBibTeX XMLCite
Full Text: arXiv